Real-rooted atlas

A038303 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(8 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(121 + 66\,x + 9\,x^{2} + 33\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114317
2187849280
324201683378274508
430613290401009815128171344
53806664591952178005049056702431114064
646704796851988413275513068002272052041272917210368
7566899529808005971945874289292856860700954261714422187263533760
86820509851360558848117696070857556699217357571032931360381704424494465614005713152
981456374781841537659518367544448105926463723885077196937308834148191120147228848266861968360507351296
10966758553312443691243402762306489251836754444807944484779023310463176468654417063510480055210815275562059049909086058496
1111412466824443190303225923403209055161030385371381751515322416960524335995414128207547468220937082302619807300202439655909354617714713594968783872
121340964851871741084880567984574254580666144838508661932027346834243575109103214021123146015972484659353101264994216870351047922920072878275829760696531441202478258483200
13156921418836050522976292229963801155171075688746530954865982471754594538370213305307099885709170891137281752780327526832143463686624308273104854086899388025838661429672226215943233004777355890688
14182970374362834365906995911141001098250213682922311216172395059632783857502609281139627385941223081493137149699195425502534682368920209671951571500028305375618094747040371560452493609043531497312487308478296944448027302232064
Technical details
Node id
n-83fdacfbb15a5bb9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bb24aeb74e5c1fb8

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         14,           3;
  2:        187,          84,           9;
  3:       2420,        1683,         378,          27;
  4:      30613,       29040,       10098,        1512,         81;
  5:     380666,      459195,      217800,       50490,       5670,       243;
  6:    4670479,     6851988,     4132755,     1306800,     227205,     20412,       729;
  7:   56689952,    98080059,    71945874,    28929285,    6860700,    954261,     71442,     2187;
  8:  682050985,  1360558848,  1176960708,   575566992,  173575710,  32931360,   3817044,   244944,   6561;
  9: 8145637478, 18415376595, 18367544448, 10592646372, 3885077196, 937308834, 148191120, 14722884, 826686, 19683;
%F Formula
The row polynomials satisfy (3*n+8+3*x)*P(n,x) = (33*n+121+(9*n+66)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((33*n + 121 + (9*n + 66)*x + 9*x^2)*P[n - 1, x])/(3*n + 8 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((33*n + 121 + (9*n + 66)*x + 9*x^2)*v[i - 1])/(3*n + 8 + 3*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.